## Someone please help me Asap! What are the zeros of the function? Show all work for credit. f(x)=3x^4−x^3−27x^2+9x

Question

What are the zeros of the function? Show all work for credit.

f(x)=3x^4−x^3−27x^2+9x

0

x = 1/3, x = -3, x = 0 and x = 3

Step-by-step explanation:

Note that x can be factored out of f(x)=3x^4−x^3−27x^2+9x immediately:

f(x) = x(3x^3 – x^2 – 27x + 9.

Let’s guess at the zeros and use synthetic div. to determine whether our guess actually is a zero:

Is 3 (a factor of 9) a zero?  Use 3 as a divisor in synth. div.:

3   /   3   -1   -27    9

9   24   -9

————————-

3      8    -3    0

Because the remainder is zero, 3 is a zero of the given polynomial.  So is 0 (which we know from having factored x out of the given f(x)=3x^4−x^3−27x^2+9x).  The quotient is 3x^2 + 8x – 3, using the coefficients derived thru synth. div., above.

Let’s use the quadratic formula to find the zeros of 3x^2 + 8x – 3:

a = 3, b = 8, c = -3

Then the discriminant is b^2 – 4(a)(c) = 8^2 – 4(3)(-3) = 64 + 36 = 100, and the square root of that is 10.

Thus, the zeros of 3x^2 + 8x – 3 are:

-8 plus or minus 10

x = ——————————-

2(3)

or x = 1/3 and x = -3